horizontal, if ψq(v1,...,vk)=0\psi_q(v_1,...,v_k)=0 whenever one of the vectors viv_i is vertical.

equivariant, if rg*ψ=ρ(g−1,ψ)r_g^{*}\psi = \rho(g^{-1},\psi) for all g∈Gg\in G, where rg:P→Pr_g: P \to P denotes the right action of GG on PP.

We denote by Ωρk(P,V)\Omega^k_{\rho}(P,V) the space of horizontal and equivariant forms. Note that Ωρk(P,V)\Omega_{\rho}^k(P,V) is in general not closed under the ordinary exterior derivative. There is a canonical isomorphism

where H:TP→TPH: TP \to TP denotes the projection to the horizontal subspace defined by ω\omega.

Some facts are:

Every form in the image of dω\mathrm{d}^{\omega} is horizontal. If a form ψ\psi is equivariant, dω\mathrm{d}^{\omega} is also equivariant.

The restriction of dω\mathrm{d}^{\omega} to Ωρk(P,V)\Omega^k_{\rho}(P,V) can be described in terms of the connection 1-form ω∈Ω1(P,𝔤)\omega\in \Omega^1(P,\mathfrak{g}) and the derivative dρ:𝔤×V→V\mathrm{d}\rho: \mathfrak{g} \times V \to V of the representation ρ\rho:

Here we have used the following general notation: if U,V,WU,V,W are vector spaces, φ∈Ωp(M,V)\varphi \in \Omega^p(M,V), ψ∈Ωq(M,W)\psi\in\Omega^q(M,W) and f:V×W→Uf: V \times W \to U is a linear map, we have φ∧fψ∈Ωp+q(M,U)\varphi \wedge_{f} \psi \in \Omega^{p+q}(M,U).

Under the isomorphism Ωρk(P,V)≅Ωk(M,P×ρV)\Omega^k_{\rho}(P,V) \cong \Omega^k(M,P \times_{\rho}V), the curvature Ω\Omega corresponds to a 2-form Fω∈Ω2(M,Ad(P))F_{\omega} \in \Omega^2(M,\mathrm{Ad}(P)), and the Bianchi-identity corresponds to DωFω=0\mathrm{D}^{\omega}F_{\omega} = 0.